Properties

Label 3.18.a.b
Level 3
Weight 18
Character orbit 3.a
Self dual Yes
Analytic conductor 5.497
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(5.49666262034\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14569}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{14569}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 297 - \beta ) q^{2} + 6561 q^{3} + ( 88258 - 594 \beta ) q^{4} + ( 191430 + 3520 \beta ) q^{5} + ( 1948617 - 6561 \beta ) q^{6} + ( 12235784 + 29376 \beta ) q^{7} + ( 65170116 - 133604 \beta ) q^{8} + 43046721 q^{9} +O(q^{10})\) \( q + ( 297 - \beta ) q^{2} + 6561 q^{3} + ( 88258 - 594 \beta ) q^{4} + ( 191430 + 3520 \beta ) q^{5} + ( 1948617 - 6561 \beta ) q^{6} + ( 12235784 + 29376 \beta ) q^{7} + ( 65170116 - 133604 \beta ) q^{8} + 43046721 q^{9} + ( -404691210 + 854010 \beta ) q^{10} + ( -493776756 - 128128 \beta ) q^{11} + ( 579060738 - 3897234 \beta ) q^{12} + ( -1259699122 + 3383424 \beta ) q^{13} + ( -217782648 - 3511112 \beta ) q^{14} + ( 1255972230 + 23094720 \beta ) q^{15} + ( 25305661960 - 26993736 \beta ) q^{16} + ( -17156563182 + 21313920 \beta ) q^{17} + ( 12784876137 - 43046721 \beta ) q^{18} + ( 40026771092 - 116754048 \beta ) q^{19} + ( -257263047540 + 196958740 \beta ) q^{20} + ( 80278978824 + 192735936 \beta ) q^{21} + ( -129851425044 + 455722740 \beta ) q^{22} + ( 148614371352 - 1365533312 \beta ) q^{23} + ( 427581131076 - 876575844 \beta ) q^{24} + ( 898347630175 + 1347667200 \beta ) q^{25} + ( -817768577538 + 2264576050 \beta ) q^{26} + 282429536481 q^{27} + ( -1208069610352 - 4675388688 \beta ) q^{28} + ( -235187034786 + 2063000384 \beta ) q^{29} + ( -2655179028810 + 5603159610 \beta ) q^{30} + ( 1700377227296 - 4379766336 \beta ) q^{31} + ( 2513249815824 - 15811058064 \beta ) q^{32} + ( -3239669296116 - 840647808 \beta ) q^{33} + ( -7890201769374 + 23486797422 \beta ) q^{34} + ( 15900669077040 + 48693407360 \beta ) q^{35} + ( 3799217502018 - 25569752274 \beta ) q^{36} + ( 5326006090214 - 58058187264 \beta ) q^{37} + ( 27196858542132 - 74702723348 \beta ) q^{38} + ( -8264885939442 + 22198644864 \beta ) q^{39} + ( -49188865789800 + 203822994600 \beta ) q^{40} + ( -56688399724374 - 64587956096 \beta ) q^{41} + ( -1428871953528 - 23036405832 \beta ) q^{42} + ( 30818515744940 - 186387709824 \beta ) q^{43} + ( -33600387667176 + 281995072040 \beta ) q^{44} + ( 8240433801030 + 151524457920 \beta ) q^{45} + ( 223188561694296 - 554177765016 \beta ) q^{46} + ( -139822820963280 + 61433939072 \beta ) q^{47} + ( 166030448119560 - 177105901896 \beta ) q^{48} + ( 30234681237945 + 718876781568 \beta ) q^{49} + ( 90101775230775 - 498090471775 \beta ) q^{50} + ( -112564211037102 + 139840629120 \beta ) q^{51} + ( -374699460462052 + 1046875513860 \beta ) q^{52} + ( -265482019305738 - 343991051712 \beta ) q^{53} + ( 83881572334857 - 282429536481 \beta ) q^{54} + ( -153660640038840 - 1762621724160 \beta ) q^{55} + ( 282790173123360 + 279687642080 \beta ) q^{56} + ( 262615645134612 - 766023308928 \beta ) q^{57} + ( -340353222681906 + 847898148834 \beta ) q^{58} + ( 863762115543228 + 2656278051328 \beta ) q^{59} + ( -1687902854909940 + 1292246293140 \beta ) q^{60} + ( 1392143828013950 + 50768356608 \beta ) q^{61} + ( 1079291378249568 - 3001167829088 \beta ) q^{62} + ( 526710380064264 + 1264540476096 \beta ) q^{63} + ( -497266784711648 - 3671011095840 \beta ) q^{64} + ( 1320461339905620 - 3786452053120 \beta ) q^{65} + ( -851955199713684 + 2989996897140 \beta ) q^{66} + ( -1664650838348092 + 534402316800 \beta ) q^{67} + ( -3174257240883036 + 12072122481468 \beta ) q^{68} + ( 975058890440472 - 8959264060032 \beta ) q^{69} + ( -1662229550569680 - 1438727091120 \beta ) q^{70} + ( -6744701103252408 - 1573308147840 \beta ) q^{71} + ( 2805359800989636 - 5751214112484 \beta ) q^{72} + ( 218294959068362 + 36563049105408 \beta ) q^{73} + ( 9194471381036502 - 22569287707622 \beta ) q^{74} + ( 5894058801578175 + 8842044499200 \beta ) q^{75} + ( 12626173834555688 - 34080380797032 \beta ) q^{76} + ( -6535270505868192 - 16072932516608 \beta ) q^{77} + ( -5365379637226818 + 14857883464050 \beta ) q^{78} + ( 2188020782607440 - 29178746268480 \beta ) q^{79} + ( -7614585847354320 + 83908519216720 \beta ) q^{80} + 1853020188851841 q^{81} + ( -8367617326875462 + 37505776763862 \beta ) q^{82} + ( -19943221132806444 - 15615881792128 \beta ) q^{83} + ( -7926144713519472 - 30675225181968 \beta ) q^{84} + ( 6553071925276140 - 56310978695040 \beta ) q^{85} + ( 33592442076079884 - 86175665562668 \beta ) q^{86} + ( -1543062135230946 + 13535345519424 \beta ) q^{87} + ( -29934904994740944 + 57620433085776 \beta ) q^{88} + ( 3486092048053722 + 113966194036992 \beta ) q^{89} + ( -17420629608022410 + 36762330201210 \beta ) q^{90} + ( -2381098286163344 + 4393923836544 \beta ) q^{91} + ( 119472162668019504 - 208796175633584 \beta ) q^{92} + ( 11156174988289056 - 28735646930496 \beta ) q^{93} + ( -49582657351153872 + 158068700867664 \beta ) q^{94} + ( -46225029707742600 + 118544006835200 \beta ) q^{95} + ( 16489432041621264 - 103736351957904 \beta ) q^{96} + ( 91784777856230498 - 48656866579200 \beta ) q^{97} + ( -85280142148308063 + 183271722887751 \beta ) q^{98} + ( -21255470251817076 - 5515490268288 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 594q^{2} + 13122q^{3} + 176516q^{4} + 382860q^{5} + 3897234q^{6} + 24471568q^{7} + 130340232q^{8} + 86093442q^{9} + O(q^{10}) \) \( 2q + 594q^{2} + 13122q^{3} + 176516q^{4} + 382860q^{5} + 3897234q^{6} + 24471568q^{7} + 130340232q^{8} + 86093442q^{9} - 809382420q^{10} - 987553512q^{11} + 1158121476q^{12} - 2519398244q^{13} - 435565296q^{14} + 2511944460q^{15} + 50611323920q^{16} - 34313126364q^{17} + 25569752274q^{18} + 80053542184q^{19} - 514526095080q^{20} + 160557957648q^{21} - 259702850088q^{22} + 297228742704q^{23} + 855162262152q^{24} + 1796695260350q^{25} - 1635537155076q^{26} + 564859072962q^{27} - 2416139220704q^{28} - 470374069572q^{29} - 5310358057620q^{30} + 3400754454592q^{31} + 5026499631648q^{32} - 6479338592232q^{33} - 15780403538748q^{34} + 31801338154080q^{35} + 7598435004036q^{36} + 10652012180428q^{37} + 54393717084264q^{38} - 16529771878884q^{39} - 98377731579600q^{40} - 113376799448748q^{41} - 2857743907056q^{42} + 61637031489880q^{43} - 67200775334352q^{44} + 16480867602060q^{45} + 446377123388592q^{46} - 279645641926560q^{47} + 332060896239120q^{48} + 60469362475890q^{49} + 180203550461550q^{50} - 225128422074204q^{51} - 749398920924104q^{52} - 530964038611476q^{53} + 167763144669714q^{54} - 307321280077680q^{55} + 565580346246720q^{56} + 525231290269224q^{57} - 680706445363812q^{58} + 1727524231086456q^{59} - 3375805709819880q^{60} + 2784287656027900q^{61} + 2158582756499136q^{62} + 1053420760128528q^{63} - 994533569423296q^{64} + 2640922679811240q^{65} - 1703910399427368q^{66} - 3329301676696184q^{67} - 6348514481766072q^{68} + 1950117780880944q^{69} - 3324459101139360q^{70} - 13489402206504816q^{71} + 5610719601979272q^{72} + 436589918136724q^{73} + 18388942762073004q^{74} + 11788117603156350q^{75} + 25252347669111376q^{76} - 13070541011736384q^{77} - 10730759274453636q^{78} + 4376041565214880q^{79} - 15229171694708640q^{80} + 3706040377703682q^{81} - 16735234653750924q^{82} - 39886442265612888q^{83} - 15852289427038944q^{84} + 13106143850552280q^{85} + 67184884152159768q^{86} - 3086124270461892q^{87} - 59869809989481888q^{88} + 6972184096107444q^{89} - 34841259216044820q^{90} - 4762196572326688q^{91} + 238944325336039008q^{92} + 22312349976578112q^{93} - 99165314702307744q^{94} - 92450059415485200q^{95} + 32978864083242528q^{96} + 183569555712460996q^{97} - 170560284296616126q^{98} - 42510940503634152q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
60.8511
−59.8511
−65.1063 6561.00 −126833. 1.46604e6 −427163. 2.28730e7 1.67913e7 4.30467e7 −9.54488e7
1.2 659.106 6561.00 303349. −1.08318e6 4.32440e6 1.59855e6 1.13549e8 4.30467e7 −7.13934e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} - 594 T_{2} - 42912 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(3))\).